Lesson 4: Electric
Fields

Lesson 4: Electric
Fields

Electric Field Lines

In the previous section of Lesson
4, the vector nature of the
electric field strength was discussed. The magnitude or
strength of an electric field in the space surrounding a
source charge is related directly to the quantity of charge
on the source charge and inversely to the distance from the
source charge. The direction of the electric field is always
directed in the direction that a positive test charge would
be pushed or pulled if placed in the space surrounding the
source charge. Since electric field is a vector quantity, it
can be represented by a vector arrow. For any given
location, the arrows point in the direction of the electric
field and their length is proportional to the strength of
the electric field at that location. Such vector arrows are
shown in the diagram below. Note that the length of the
arrows are longer when closer to the source charge and
shorter when further from the source charge.

A more useful means of visually
representing the vector nature of an electric field is
through the use of electric field lines of force. Rather
than draw countless vector arrows in the space surrounding a
source charge, it is perhaps more useful to draw a pattern
of several lines which extend between infinity and
the source charge. These pattern of lines, sometimes
referred to as electric field
lines, point in the direction which a positive
test charge would accelerate if placed upon the line. As
such, the lines are directed away from positively charged
source charges and toward negatively charged source charges.
To communicate information about the direction of the field,
each line must include an arrowhead which points in the
appropriate direction. An electric field line pattern could
include an infinite number of lines. Because drawing such
large quantities of lines tends to decrease the readability
of the patterns, the number of lines are usually limited.
The presence of a few lines around a charge is typically
sufficient to convey the nature of the electric field in the
space surrounding the lines.

Rules for Drawing
Electric Field Patterns

There are a variety of conventions and rules to drawing
such patterns of electric field lines. The conventions are
simply established in order that electric field line
patterns communicate the greatest amount of information
about the nature of the electric field surrounding a charged
object. One common convention is to surround more charged
objects by more lines. Objects with greater charge create
stronger electric fields. By surrounding a highly charged
object with more lines, one can communicate the strength of
an electric field in the space surrounding a charged object
by the line density. This convention is depicted in the
diagram below.

Not only does the density of lines
surrounding any given object reveal information about the
quantity of charge on the source charge, the density of
lines at a specific location in space reveals information
about
the strength of the field at that location. Consider the
object shown at the right. Two different circular
cross-sections are drawn at different distances from the
source charge. These cross-sections represent regions of
space closer to and further from the source charge. The
field lines are closer together in the regions of space
closest to the charge; and they are spread further apart in
the regions of space furthest from the charge. Based on the
convention concerning line density, one would reason that
the electric field is greatest at locations closest to the
surface of the charge and least at locations further from
the surface of the charge. Line density in an electric field
line pattern reveals information about the strength or
magnitude of an electric field.

A second rule for drawing electric field
lines involves drawing the lines of force perpendicular to
the surfaces of objects at the locations where the lines
connect to object's surfaces. At the surface of both
symmetrically shaped and irregularly shaped objects, there
is never a component of electric force which is directed
parallel to the surface. The electric force, and thus the
electric field, is always directed perpendicular to the
surface of an object. If there were ever any component of
force parallel to the surface, then any excess charge
residing upon the surface of a source charge would begin to
accelerate. This would lead to the occurrence of an electric
current within the object; this is never observed in
static electricity. Once a line of force leaves the
surface of an object, it will often alter its direction.
This occurs when drawing electric field lines for
configurations of two or more charges as discussed in the
section below.

A final rule for drawing electric field
lines involves the intersection of lines. Electric field
lines should never cross. This is particularly important
(and tempting to break) when drawing electric field lines
for situations involving a configuration of charges (as in
the section below). If electric field
lines were ever allowed to cross each other at a given
location, then you might be able to imagine the results.
Electric field lines reveal information about the direction
(and the strength) of an electric field within a region of
space. If the lines cross each other at a given location,
then there must be two distinctly different values of
electric field with their own individual direction at that
given location. This could never be the case. Every single
location in space has its own electric field strength and
direction associated with it. Consequently, the lines
representing the field cannot cross each other at any given
location in space.

Electric
Field Lines for Configurations of Two or More
Charges

In the examples above, we've seen electric field lines
for the space surrounding single point charges. But what if
a region of space contains more than one point charge? How
can the electric field in the space surrounding a
configuration of two or more charges be described by
electric field lines? To answer this question, we will first
return to our original method of drawing electric field
vectors.

Suppose that there are two positive charges - charge A
(QA) and charge B (QB) - in a given
region of space. Each charge creates its own electric field.
At any given location surrounding the charges, the strength
of the electric field can be calculated using the expression
kQ/d2. Since there are two charges, the
kQ/d2 calculation would have to be performed
twice at each location - once with
kQA/dA2 and once with
kQB/dB2 (dA is
the distance from that location to the center of charge A
and dB is the distance from that location to the
center of charge B). The results of these calculations are
illustrated in the diagram below with electric field vectors
(EA and EB) drawn at a variety of
locations. The strength of the field is represented by the
length of the arrow and the direction of the field is
represented by the direction of the arrow.

Since electric field is a vector, the
usual operations which apply to vectors can be applied to
electric field. That is, they can be added in head-to-tail
fashion to determine the resultant or net electric field
vector at each location. This is shown in the diagram
below.

The diagram above shows that the magnitude
and direction of the electric field at each location is
simply the vector sum of the electric field vectors for each
individual charge. If more locations are selected and the
process of drawing EA, EB and
Enet is repeated, then the electric field
strength and direction at a multitude of locations will be
known. (This is not done since it is a highly time intensive
task.) Ultimately, the electric field lines surrounding the
configuration of our two charges would begin to emerge. For
the limited number of points selected in this location, the
beginnings of the electric field line pattern can be seen.
This is depicted in the diagram below. Note that for each
location, the electric field vectors point tangent to the
direction of the electric field lines at any given
point.

The construction of electric field lines
in this manner is a tedious and cumbersome task. The use of
a field plotting computer software program or a lab
procedure produces similar results in less time (and with
more phun). Whatever the method used to determine the
electric field line patterns for a configuration of charges,
the general idea is that the pattern is the resultant of the
patterns for the individual charges within the
configuration. The electric field line patterns for other
charge configurations are shown in the diagrams below.

In each of the above diagrams, the
individual source charges in the configuration possess the
same amount of charge. Having an identical quantity of
charge, each source charge has an equal ability to alter the
space surrounding it. Subsequently, the pattern is
symmetrical in nature and the number of lines emanating from
a source charge or extending towards a source charge are the
same. This reinforces a principle discussed
earlier which stated that the density of lines
surrounding any given source charge is proportional to the
quantity of charge on that source charge. If the quantity of
charge on a source charge is not identical, the pattern will
take on an asymmetric nature as one of the source charges
will have a greater ability to alter the electrical nature
of the surrounding space. This is depicted in the electric
field line patterns below.

After plotting the electric field line
patterns for a variety of charge configurations, the general
patterns for other configurations can be predicted. There
are a number of principles which will assist in such
predictions. These principles are described (or
re-described) in the list below.

Electric field lines always extend from a positively
charged object to a negatively charged object, from a
positively charged object to infinity, or from infinity
to a negatively charged object.

Electric field lines never cross each other.

Electric field lines are most dense around objects
with the greatest amount of charge.

At locations where electric field lines meet the
surface of an object, the lines are perpendicular to the
surface.

Electric Field
Lines as an Invisible Reality

It has been emphasized in Lesson 4 that the concept of an
electric field arose as scientists attempted to explain the
action-at-a-distance which occurs between charged objects.
The concept of the electric field was first introduced by
19th century physicist Michael Faraday. It was Faraday's
perception that the pattern of lines characterizing the
electric field represent an invisible reality. Rather than
thinking in terms of one charge affecting another charge,
Faraday used the concept of a field to propose that a
charged object (or a massive object in the case of a
gravitational field) affects the space that surrounds it. As
another object enters that space, it becomes effected by the
field established in that space. Viewed in this manner, a
charge is seen to interact with an electric field as opposed
to with another charge. To Faraday, the secret to
understanding action-at-a-distance is to understand the
power of charge-field-charge. A charged object sends its
electric field into space, reaching from the "puller to the
pullee." Each charge or configuration of charges creates an
intricate web of influence in the space surrounding it.
While the lines are invisible, the affect is ever so real.
So as you practice the exercise of constructing electric
field lines around charges or configuration of charges, you
are doing more than simply drawing curvy lines. Rather, you
are describing the electrified web of space that will draw
and repel other charges which enter it.

Check
Your Understanding

Use your understanding to answer the following questions.
When finished, click the button to view the answers.

1. Several electric field line patterns are shown in the
diagrams below. Which of these patterns are incorrect?
_________ Explain what is wrong with all incorrect
diagrams.

2. Erin Agin drew the following electric field lines for
a configuration of two charges. What did Erin do wrong?
Explain.

3. Consider the electric field lines shown in the diagram
below. From the diagram, it is apparent that object A is
____ and object B is ____.

a. +, +

b. -, -

c. +, -

d. -, +

e. insufficient info

4.
Consider the electric field lines drawn at the right for a
configuration of two charges. Several locations are labeled
on the diagram. Rank these locations in order of the
electric field strength - from smallest to largest.

5. Use your understanding of electric field lines to
identify the charges on the objects in the following
configurations.

6. Observe the electric field lines below for various
configurations. Rank the objects according to which has the
greatest magnitude of electric charge, beginning with the
smallest charge.